OP how long have you been investigating the London private school sector?
I don't mean to sound off, but I will tell you that it is incredibly hard to get into the selective schools, not because your DS is not clever enough, there just seems to be too many applicants for the spaces.
So you may well find that Croydon and Emmanuel may well feature on your list soon enough, unless you are fine with less academic schools, and it does not sound like you are. Also on closer inspection, you will find schools like Whitgift offer facilities matched by few and are incredibly rare. So called 'back up' schools no longer exist in London nowadays from the looks of it. Infact the schools you mention are incredibly hard to get into. One day you should go to Clapham Junction and see the hoards of DC going to Whitgift, you would be shocked at how far they come from.
I agree with you that getting in at 10+ is a better option. Taking them to a prep till 13+ even better (the schools have relationships, and this is what prep schools prepare for, and may help your DS win a scholarship, you may not have expected). I know parents who were tearing their hair out at 11+ from state and their DC did not know a lot of things, even science lab rules etc. If your DC is not challenged I can see why you need to move him. I also think schools tend to be more lenient with state students admissions at 10+
Co-incidentally, I know someone who moved their DS from City to one of the top selective after one year - they had no issues with city, just how things panned out.
The not very central schools have coaches, but it depends on where you are. From your selection I assume, closer to the East.
For anyone else, do ring up the school if you just marginally missed the deadline, some even selective may be accommodating.